arXiv subjects

Patrick Bieker

Publications and source records attributed to Patrick Bieker.

7 recordsLinked to original sources

arXiv2026

A higher Gross-Zagier type formula for moduli of shtukas at deeper level

We generalize the function field analogue analogue of the Gross-Zagier and Waldspurger formulae due to Yun and Zhang relating the (higher) derivative of an automorphic (base change) L-function to intersection numbers of Heegner-Drinfeld cycles on moduli spaces of shtukas to not necessarily square-free level. The main new input is the study of the geometry and cohomology of the appropriate ``integral models'' of moduli spaces of shtukas at deeper level.

math.NT↗

arXiv2025

Normality of Schubert varieties in affine Grassmannians II: The tamely ramified case

We prove a criterion for the normality of Schubert varieties in twisted affine Grassmannians in terms of the order of the algebraic fundamental group of a certain Levi subgroup, in particular in small positive characteristic. As an application, we obtain a similar normality criterion for local models in both equal and mixed characteristic. In particular, we give a classification of normal Pappas-Zhu local models at absolutely special level as well as for adjoint groups of rank 1.

math.AG↗

arXiv2024

A converse theorem for Borcherds products in signature $(2,2)$

We show that a modular unit on two copies of the upper half-plane is a Borcherds product if and only if its boundary divisor is a special boundary divisor. Therefore, we define a subspace of the space of invariant vectors for the Weil representation which maps surjectively onto the space of modular units that are Borcherds products. Moreover, we show that every boundary divisor of a Borcherds product can be obtained in this way. As a byproduct we obtain new identities of eta products.

math.NT↗

arXiv2024

Integral models of moduli spaces of shtukas with deep Bruhat-Tits level structures

We construct integral models for moduli spaces of shtukas with deep Bruhat-Tits level structures. We embed the moduli space of global shtukas for a deep Bruhat-Tits group scheme into the limit of the moduli spaces of shtukas for all associated parahoric group schemes. Its schematic image defines an integral model of the moduli space of shtukas with deep Bruhat-Tits level with favourable properties: They admit proper, surjective and generically étale level maps as well as a natural Newton stratification. In the Drinfeld case, this general construction of integral models recovers the moduli space of Drinfeld shtukas with Drinfeld level structures.

math.AG↗

arXiv2023

Normality of Schubert varieties in affine Grassmannians

We classify all normal Schubert varieties in the affine Grassmannian of a semisimple group over an arbitrary field with special attention to small positive characteristic. The proof is elementary and relies on tangent space calculations for quasi-minuscule Schubert varieties, a refined Levi lemma in positive characteristic and the classification of minimal degenerations.

math.AG↗

arXiv2023

Compactification of Level Maps of Moduli Spaces of Drinfeld Shtukas

We define Drinfeld level structures for Drinfeld shtukas of any rank and show that their moduli spaces are regular and admit finite flat level maps. In particular, the moduli spaces of Drinfeld shtukas with Drinfeld $Γ_0(\mathfrak{p}^n)$-level structures provide a good integral model and relative compactification of the moduli space of shtukas with naive $Γ_0(\mathfrak{p}^n)$-level defined using shtukas for dilated group schemes.

math.AG↗

arXiv2023

Invariants for the Weil representation and Modular Units for Orthogonal Groups of Signature (2,2)

We show that the space of invariants for the Weil representation for discriminant groups which contain self-dual isotropic subgroups is spanned by the characteristic functions of the self-dual isotropic subgroups. As an application, we construct modular units for orthogonal groups in signature (2,2) using Borcherds products.

math.NT↗